Can solutions of GR have non-zero genus? Imagine there is "cavity" in one's locally Lorentzian manifold (the manifold has non-zero genus). Have these kind of solutions in general relativity been considered? Is there any experiment one can do to prove we live in a non-zero genus world?
 A: Any manifold, regardless of its properties, is a solution to the Einstein field equations for some stress-energy tensor $T_{\mu\nu}$, providing it admits a metric, $g_{\mu\nu}$.
In theory, you are free to pick your favourite algebraic manifold of genus $g$, compute the metric and find its corresponding stress-energy tensor.
However, if we restrict the stress-energy, say by requiring that $T_{\mu\nu}$ be infinitely differentiable, then we also restrict the space of allowable metrics. Likewise, if we demand the manifold be globally hyperbolic, which is a frequent requirement due to the implication on causality, then we also restrict possible solutions.
Based on a cursory search, I cannot find any classification of genus $g$ spacetime solutions yet. However, in the paper here it is written that,

In the current paper we study that equation on closed 2-dimensional
  surfaces that have genus $>0$. We derive all the solutions assuming
  the embeddability in 4-dimensional spacetime that satisfies the vacuum
  Einstein equations...

It seems we can have sub-manifolds in spacetime which do have non-zero genus, and are sensible solutions to the vacuum Einstein field equations, meaning at least $T_{\mu\nu}$ is physically sensible, being zero.
